Limit distributions of some integral functionals for null-recurrent diffusions.
It is well known that if \((X,\mathbf P_ {x})\) is an ergodic positive recurrent time-homogeneous Markov process on \(\mathbb R\) with an invariant probability measure \(\mu \), then under suitable hypotheses \[ T^ {-1}\int ^ {T}_ 0 f(X(t))\, dt \to \mu (f)\text{ as }T\to \infty\;\mathbf P_ {x}\text{-almost surely} \] for all \(\mu \)-integrable functions \(f\), and, moreover, a central limit theorem holds. Recently, the author has devoted several papers to an analogous problem of the long-time behaviour of the functional \(T^ {-1}\int ^ {T} _ {0} f(X(t))dt\) for null-recurrent diffusions. The paper under review contains another result in this direction: It is supposed that \((X,\mathbf P_ {x})\) is a diffusion process with the generator \(L = a(x)\, d^ 2/ dx^ {2}\), \(x\in \mathbb R\), where \(a\) is a strictly positive locally Lipschitz continuous function. There exist constants \(p_ +\), \(p_ -\) and \(\beta >-1\) such that \(p_ + + p_ - >0\) and the limits \(\lim _ {X\to \pm \infty } X^ {-1}\int ^ {X}_ 0 | x | ^ {-\beta } a(x)^ {-1}dx = p_ \pm \) exist. Moreover, \(\sup _ {| x| >\varepsilon } a(x)^ {-1}| x| ^ {-\beta } <\infty \) for all \(\varepsilon >0\). The function \(f\) is piecewise continuous, bounded, and \[ \lim _ {X\to \pm \infty } \left [\int ^ {X}_ 0 | x| ^ {-\beta } a(x)^ {-1}dx\right ]^ {-1}\int ^ {X}_ {0} f(x)| x | ^ {-\beta }a(x)^ {-1}\, dx = f_ \pm \] for some constants \(f_ +\), \(f_ -\) such that \(f_ +\neq f_ -\). Then it is proven that \[ \lim _ {T\to \infty } \mathbf P_ {x}\left \{ {\int ^ {T}_ 0 (f(X(t)) -f_ -)\, dt \over T(f_ + - f_ -)} < x\right \} = F_ {\alpha ,A}(x), \] where \(\alpha (2+\beta ) = 1\), \(A = (p_ {+}/p_ {-})^ \alpha \) and \(F_ {\alpha ,A}\) is the generalized arc-sine law introduced by \textit{J. Lamperti} [Trans. Am. Math. Soc. 88, 380-387 (1958; Zbl 0228.60046)]. Finally, it is shown how to extend this result to diffusions with generators of the form \(L_ 1 = a(x)\, d^ 2 / dx^ 2 + b(x)\, d/ dx\).
scientific article; zbMATH DE number 2102373